Interpolation Operators

This page covers the theory behind the interpolation (averaging) operators in finitevolX, explains why staggered-grid interpolation is necessary, and gives practical guidance for composing operators.

The Problem

On an Arakawa C-grid, scalars live at T-points (cell centres) while velocities live at U/V-points (face centres) and vorticity at X-points (corners). Many operations require a quantity at a location other than where it is naturally defined:

Computing kinetic energy requires u² and v² both at T-points, but they are stored at U- and V-points respectively.
The Coriolis term \((f\,v)\) at U-points requires v to be interpolated from V-points to U-points (a cross-face average).
The potential vorticity \(((\zeta + f)/h)\) requires h to be interpolated to X-points where \(\zeta\) is defined.

finitevolX provides a complete set of second-order linear (arithmetic) averaging operators that move quantities between all four location types.

Staggered Grid Locations

All arrays share the same shape [Ny, Nx] with a one-cell ghost ring.

Symbol	Position	Stored value
T	`(j, i)` — cell centre	Scalars: \(h\), \(T\), \(p\)
U	`(j, i+½)` — east face	x-velocity \(u\)
V	`(j+½, i)` — north face	y-velocity \(v\)
X	`(j+½, i+½)` — NE corner	Vorticity \(\zeta\), PV \(q\)

Averaging Stencils

1-D (2-point) averages

For 1-D arrays or for a single direction in 2-D:

\[ \text{T} \to \text{U}: \quad h_{i+\frac{1}{2}} = \tfrac{1}{2}(h_i + h_{i+1}) \]

\[ \text{U} \to \text{T}: \quad u_i = \tfrac{1}{2}(u_{i+\frac{1}{2}} + u_{i-\frac{1}{2}}) \]

These are the building blocks for all 2-D averages.

Face-to-face averages (bilinear 4-point)

Moving between the two face types (U ↔ V) requires a 4-point bilinear average because the target point is displaced in both directions:

\[ u_{j+\frac{1}{2},i} = \tfrac{1}{4} \bigl(u_{j,i+\frac{1}{2}} + u_{j+1,i+\frac{1}{2}} + u_{j,i-\frac{1}{2}} + u_{j+1,i-\frac{1}{2}}\bigr) \]

\[ v_{j, i+\frac{1}{2}} = \tfrac{1}{4} \bigl(v_{j+\frac{1}{2},i} + v_{j-\frac{1}{2},i} + v_{j+\frac{1}{2},i+1} + v_{j-\frac{1}{2},i+1}\bigr) \]

These cross-face averages are used in the Coriolis operator (interpolating the cross-component velocity to the target face) and in the energy-conserving momentum advection scheme.

T ↔ X (corner) averages (bilinear 4-point)

\[ h_{j+\frac{1}{2}, i+\frac{1}{2}} = \tfrac{1}{4} \bigl(h_{j,i} + h_{j,i+1} + h_{j+1,i} + h_{j+1,i+1}\bigr) \quad (\text{T} \to \text{X}) \]

\[ q_{j,i} = \tfrac{1}{4} \bigl(q_{j+\frac{1}{2},i+\frac{1}{2}} + q_{j-\frac{1}{2},i+\frac{1}{2}} + q_{j+\frac{1}{2},i-\frac{1}{2}} + q_{j-\frac{1}{2},i-\frac{1}{2}}\bigr) \quad (\text{X} \to \text{T}) \]

Complete Operator Map

The table below lists every averaging operator in Interpolation2D and its physical interpretation.

Method	From → To	Average	Typical use
`T_to_U`	T → U	½(T_i + T_{i+1})	Height at east face (pressure gradient)
`T_to_V`	T → V	½(T_j + T_{j+1})	Height at north face (pressure gradient)
`T_to_X`	T → X	¼(T_{j,i} + T_{j,i+1} + T_{j+1,i} + T_{j+1,i+1})	h at corners for PV = (ζ+f)/h
`U_to_T`	U → T	½(U_i + U_{i-1})	u² at T-points for kinetic energy
`V_to_T`	V → T	½(V_j + V_{j-1})	v² at T-points for kinetic energy
`X_to_T`	X → T	¼(X_{j+½,i+½} + X_{j-½,i+½} + X_{j+½,i-½} + X_{j-½,i-½})	Vorticity from corner to centre
`U_to_X`	U → X	½(U_j + U_{j+1})	u at corners for vorticity-flux
`V_to_X`	V → X	½(V_i + V_{i+1})	v at corners for vorticity-flux
`X_to_U`	X → U	½(X_j + X_{j-1})	Vorticity at face for Coriolis
`X_to_V`	X → V	½(X_i + X_{i-1})	Vorticity at face for Coriolis
`U_to_V`	U → V	¼(4-point bilinear)	Cross-velocity for Coriolis term
`V_to_U`	V → U	¼(4-point bilinear)	Cross-velocity for Coriolis term

Ghost-Cell Conventions

All operators write only to [1:-1, 1:-1]. The ghost ring provides the stencil data for the last interior cell:

Direction	Ghost consumed	Slice
T → U (x-forward)	East T ghost `T[:, Nx−1]`	`h[1:-1, 2:]`
T → V (y-forward)	North T ghost `T[Ny−1, :]`	`h[2:, 1:-1]`
U → T (x-backward)	West U ghost `U[:, 0]`	`u[1:-1, :-2]`
V → T (y-backward)	South V ghost `V[0, :]`	`v[:-2, 1:-1]`

Bilinear averages

T_to_X, U_to_V, V_to_U, and X_to_T each read ghost cells in both directions. Make sure the appropriate BCs are applied before calling these operators in a boundary region.

3-D Extension

Interpolation3D applies the same 2-D horizontal stencils to each z-level independently. Array shapes are [Nz, Ny, Nx] and only the interior [1:-1, 1:-1, 1:-1] is written. Currently T_to_U, T_to_V, U_to_T, and V_to_T are provided.

Quick Usage

import jax.numpy as jnp
from finitevolx import ArakawaCGrid2D, Interpolation2D

grid = ArakawaCGrid2D.from_interior(64, 64, 1e6, 1e6)
interp = Interpolation2D(grid=grid)

h = jnp.ones((grid.Ny, grid.Nx))   # T-point thickness
u = jnp.zeros((grid.Ny, grid.Nx))  # U-point velocity
v = jnp.zeros((grid.Ny, grid.Nx))  # V-point velocity
f = jnp.ones((grid.Ny, grid.Nx))   # T-point Coriolis parameter

# Height at east faces (for shallow-water pressure gradient)
h_u = interp.T_to_U(h)   # shape [Ny, Nx], values at U-points

# Height at NE corners (for potential vorticity)
h_q = interp.T_to_X(h)   # shape [Ny, Nx], values at X-points

# Cross-face velocity average (for Coriolis term)
v_on_u = interp.V_to_U(v)   # v interpolated to U-points
u_on_v = interp.U_to_V(u)   # u interpolated to V-points

# Coriolis parameter at velocity points
f_on_u = interp.T_to_U(f)   # f at U-points (x-average)
f_on_v = interp.T_to_V(f)   # f at V-points (y-average)

Decision Guide

What quantity do you need and where?
│
├── Scalar from T-points to...
│   ├── U-points (east face)   → T_to_U
│   ├── V-points (north face)  → T_to_V
│   └── X-points (NE corner)   → T_to_X
│
├── Velocity from face to...
│   ├── U → T  (x-mean)        → U_to_T
│   ├── V → T  (y-mean)        → V_to_T
│   ├── U → X  (y-mean)        → U_to_X
│   ├── V → X  (x-mean)        → V_to_X
│   ├── X → T  (bilinear)      → X_to_T
│   ├── X → U  (y-mean)        → X_to_U
│   └── X → V  (x-mean)        → X_to_V
│
└── Cross-face velocity for Coriolis...
    ├── u at V-points           → U_to_V  (bilinear 4-pt)
    └── v at U-points           → V_to_U  (bilinear 4-pt)

References

Arakawa & Lamb (1977) — Staggered C-grid layout
Sadourny (1975) — Cross-face averaging in the shallow-water system
Veros ocean model — C-grid interpolation conventions